| L(s) = 1 | + (4.89 + 2.82i)2-s + (15.9 + 27.7i)4-s + (36.7 − 21.2i)5-s + (−194.5 + 336. i)7-s + 181. i·8-s + 240·10-s + (1.80e3 + 1.03e3i)11-s + (−707.5 − 1.22e3i)13-s + (−1.90e3 + 1.10e3i)14-s + (−512. + 886. i)16-s − 2.36e3i·17-s − 3.06e3·19-s + (1.17e3 + 678. i)20-s + (5.87e3 + 1.01e4i)22-s + (−1.81e4 + 1.04e4i)23-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.293 − 0.169i)5-s + (−0.567 + 0.982i)7-s + 0.353i·8-s + 0.239·10-s + (1.35 + 0.780i)11-s + (−0.322 − 0.557i)13-s + (−0.694 + 0.400i)14-s + (−0.125 + 0.216i)16-s − 0.481i·17-s − 0.447·19-s + (0.146 + 0.0848i)20-s + (0.552 + 0.956i)22-s + (−1.48 + 0.860i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.181680062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.181680062\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4.89 - 2.82i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-36.7 + 21.2i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (194.5 - 336. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-1.80e3 - 1.03e3i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (707.5 + 1.22e3i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + 2.36e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 3.06e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (1.81e4 - 1.04e4i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-1.12e4 - 6.47e3i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-5.66e3 - 9.81e3i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 - 4.71e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (5.36e3 - 3.09e3i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (7.25e4 - 1.25e5i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (1.56e5 + 9.00e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 - 2.65e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (3.13e5 - 1.80e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-1.75e5 + 3.03e5i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (6.01e4 + 1.04e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 3.35e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 1.75e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-1.26e5 + 2.18e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-2.93e5 - 1.69e5i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 + 7.89e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-6.48e5 + 1.12e6i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.22409454803914600276587355086, −11.57268767712380382739250158099, −9.870605863674340052557286944176, −9.187305558026705070884868334635, −7.87306648449285120418418752146, −6.59109296226721769883920424313, −5.78155084453566623193421527522, −4.54805280720455047325942465188, −3.17640183575700694616864634103, −1.77142665621155530444265273022,
0.48689087649009089066923298043, 1.98862649485580996682960395134, 3.60018349874754483592120738510, 4.36704602584447569112308818743, 6.19352439112723331060588994263, 6.64968907469612463304456867717, 8.269889735291045762230086643138, 9.652587484368468834606705636991, 10.37508450437366391118263641409, 11.48659306503306108081805048339
